Wednesday, August 14, 2013

Trying to Understand Stuff

In the weeks just before the Fall semester, I have been trying to understand some mathematics in order to revise my book Doing Mathematics (2003). But what I am trying to understand is highly technical in the case of what are called "large cardinals" (essentially numbers much bigger than the size of the set of the integers, aleph-nought--a countable infinity--for these are uncountable infinities, such as the size of the interval 0,1 of the real line).  In the years since the mid-60s, this field of set theory has flourished, its results important but not at all easily mastered (at least by me).

I am also trying to understand the relationship between the Yang-Baxter Equations (and commuting transfer matrices), YBE,  the Bethe Ansatz, BA, and Hopf algebras, HA--in statistical mechanics and other realms. The YBE come up naturally in the solution to a model of a ferromagnet. The BA says that when you have many particles interacting with each other, you can treat the situation as a suitable sum of two-body interactions. (The consistency conditions for this to work is related to the YBE, I believe). I am not sure how the HA play a role, but they are ways of expressing the YBE I believe. I have no intuition of just what makes a HA special.

Moreover, I am not sure that my exposition of a crucial analogy in chapter 5 is really right.

I will work it out, time will help. But what's interesting for me is that when I am ready, as I now am, I become willing to work hard to figure things out, or at least to make a serious guess as to what is going on.

Time disappears when I am working, although a daily nap or two is needed to clear the mind.

It is time like these where I wished I really had a much deeper training in mathematics, training deep enough so that I had some useful intuitions about what is going on.